# Why are there 2 pi radians in a circle?

Aug 30, 2015

It is not arbitrary, but it is because of the definition of radian measure.

#### Explanation:

One definition says the radian measure of angle $\theta$ is the ratio of arc length $s$ to radius $r$ when $\theta$ is made central in a circle of radius $r$.

The arc length of the whole circle is its circumference $2 \pi r$, so the angle that goes once around the circle has radian measure $\frac{C}{r} = \frac{2 \pi r}{r} = 2 \pi$

This definition is the reason that we have the formula for arc length $s = r \theta$ as long as we measure $\theta$ in radians

Note that whatever unit are used to measure $r$ and $s$, in the definition of radian measure they cancel. We say that radian measure is dimesionless.

Also note that some teachers introduce radian measure without discussing the 'official' definition as a ratio of lengths. They simply announce that once around the circle is $2 \pi$ radians. (Some say ${360}^{\circ}$ is the same as $2 \pi$ radians.)
This makes it look arbitrary when it is not arbitrary at all.